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1 2 + 1 4 + 1 8 + 1 16 + 1 32 + ...

You might think that if you keep adding more and more terms you will eventually get larger and larger numbers, but in fact you won't even get past 1 - try it and see for yourself!

We denote the sum of an infinite number of terms of a sequence by

S = i = 1 a i

When we sum the terms of a series, and the answer we get after each summation gets closer and closer to some number, we say that the series converges . If a series does not converge, then we say that it diverges .

Infinite geometric series

There is a simple test for knowing instantly which geometric series converges and which diverges. When r , the common ratio, is strictly between -1 and 1, i.e. - 1 < r < 1 , the infinite series will converge, otherwise it will diverge. There is also a formula for working out the value to which the series converges.

Let's start off with formula [link] for the finite geometric series:

S n = i = 1 n a 1 · r i - 1 = a 1 ( r n - 1 ) r - 1

Now we will investigate the behaviour of r n for - 1 < r < 1 as n becomes larger.

Take r = 1 2 :

n = 1 : r n = r 1 = ( 1 2 ) 1 = 1 2 n = 2 : r n = r 2 = ( 1 2 ) 2 = 1 2 · 1 2 = 1 4 < 1 2 n = 3 : r n = r 3 = ( 1 2 ) 3 = 1 2 · 1 2 · 1 2 = 1 8 < 1 4

Since r is in the range - 1 < r < 1 , we see that r n gets closer to 0 as n gets larger.

Therefore,

S n = a 1 ( r n - 1 ) r - 1 S = a 1 ( 0 - 1 ) r - 1 for - 1 < r < 1 = - a 1 r - 1 = a 1 1 - r

The sum of an infinite geometric series is given by the formula

S = i = 1 a 1 r i - 1 = a 1 1 - r for - 1 < r < 1

where a 1 is the first term of the series and r is the common ratio.

Khan academy video on series - 2

Exercises

  1. What does ( 2 5 ) n approach as n tends towards ?
  2. Given the geometric series:
    2 · ( 5 ) 5 + 2 · ( 5 ) 4 + 2 · ( 5 ) 3 + ...
    1. Show that the series converges
    2. Calculate the sum to infinity of the series
    3. Calculate the sum of the first 8 terms of the series, correct to two decimal places.
    4. Determine n = 9 2 · 5 6 - n correct to two decimal places using previously calculated results.
  3. Find the sum to infinity of the geometric series 3 + 1 + 1 3 + 1 9 + ...
  4. Determine for which values of x , the geometric series
    2 + 2 3 ( x + 1 ) + 2 9 ( x + 1 ) 2 + ...
    will converge.
  5. The sum to infinity of a geometric series with positive terms is 4 1 6 and the sum of the first two terms is 2 2 3 . Find a , the first term, and r , the common ratio between consecutive terms.

End of chapter exercises

  1. Is 1 + 2 + 3 + 4 + . . . an example of a finite series or an infinite series ?
  2. Calculate
    k = 2 6 3 ( 1 3 ) k + 2
  3. If x + 1 ; x - 1 ; 2 x - 5 are the first 3 terms of a convergent geometric series, calculate the:
    1. Value of x .
    2. Sum to infinity of the series.
  4. Write the sum of the first 20 terms of the series 6 + 3 + 3 2 + 3 4 + . . . in ,-notation.
  5. Given the geometric series: 2 · 5 5 + 2 · 5 4 + 2 · 5 3 + ...
    1. Show that the series converges.
    2. Calculate the sum of the first 8 terms of the series, correct to TWO decimal places.
    3. Calculate the sum to infinity of the series.
    4. Use your answer to [link] above to determine
      n = 9 2 · 5 ( 6 - n )
      correct to TWO decimal places.
  6. For the geometric series,
    54 + 18 + 6 + . . . + 5 ( 1 3 ) n - 1
    calculate the smallest value of n for which the sum of the first n terms is greater than 80 . 99 .
  7. Determine the value of k = 1 12 ( 1 5 ) k - 1 .
  8. A new soccer competition requires each of 8 teams to play every other team once.
    1. Calculate the total number of matches to be played in the competition.
    2. If each of n teams played each other once, determine a formula for the total number of matches in terms of n .
  9. The midpoints of the opposite sides of square of length 4 units are joined to form 4 new smaller squares. This midpoints of the new smaller squares are then joined in the same way to make even smaller squares. This process is repeated indefinitely. Calculate the sum of the areas of all the squares so formed.
  10. Thembi worked part-time to buy a Mathematics book which cost R29,50. On 1 February she saved R1,60, and saves everyday 30 cents more than she saved the previous day. (So, on the second day, she saved R1,90, and so on.) After how many days did she have enough money to buy the book?
  11. Consider the geometric series:
    5 + 2 1 2 + 1 1 4 + ...
    1. If A is the sum to infinity and B is the sum of the first n terms, write down the value of:
      1. A
      2. B in terms of n .
    2. For which values of n is ( A - B ) < 1 24 ?
  12. A certain plant reaches a height of 118 mm after one year under ideal conditions in a greenhouse. During the next year, the height increases by 12 mm. In each successive year, the height increases by 5 8 of the previous year's growth. Show that the plant will never reach a height of more than 150 mm.
  13. Calculate the value of n if a = 1 n ( 20 - 4 a ) = - 20 .
  14. Michael saved R400 during the first month of his working life. In each subsequent month, he saved 10% more than what he had saved in the previous month.
    1. How much did he save in the 7 th working month?
    2. How much did he save all together in his first 12 working months?
    3. In which month of his working life did he save more than R1,500 for the first time?
  15. A man was injured in an accident at work. He receives a disability grant of R4,800 in the first year. This grant increases with a fixed amount each year.
    1. What is the annual increase if, over 20 years, he would have received a total of R143,500?
    2. His initial annual expenditure is R2,600 and increases at a rate of R400 per year. After how many years does his expenses exceed his income?
  16. The Cape Town High School wants to build a school hall and is busy with fundraising. Mr. Manuel, an ex-learner of the school and a successful politician, offers to donate money to the school. Having enjoyed mathematics at school, he decides to donate an amount of money on the following basis. He sets a mathematical quiz with 20 questions. For the correct answer to the first question (any learner may answer), the school will receive 1 cent, for a correct answer to the second question, the school will receive 2 cents, and so on. The donations 1, 2, 4, ... form a geometric sequence. Calculate (Give your answer to the nearest Rand)
    1. The amount of money that the school will receive for the correct answer to the 20 th question.
    2. The total amount of money that the school will receive if all 20 questions are answered correctly.
  17. The first term of a geometric sequence is 9, and the ratio of the sum of the first eight terms to the sum of the first four terms is 97 : 81 . Find the first three terms of the sequence, if it is given that all the terms are positive.
  18. ( k - 4 ) ; ( k + 1 ) ; m ; 5 k is a set of numbers, the first three of which form an arithmetic sequence, and the last three a geometric sequence. Find k and m if both are positive.
  19. Given: The sequence 6 + p ; 10 + p ; 15 + p is geometric.
    1. Determine p .
    2. Show that the common ratio is 5 4 .
    3. Determine the 10 th term of this sequence correct to one decimal place.
  20. The second and fourth terms of a convergent geometric series are 36 and 16, respectively. Find the sum to infinity of this series, if all its terms are positive.
  21. Evaluate: k = 2 5 k ( k + 1 ) 2
  22. S n = 4 n 2 + 1 represents the sum of the first n terms of a particular series. Find the second term.
  23. Find p if:        k = 1 27 p k = t = 1 12 ( 24 - 3 t )
  24. Find the integer that is the closest approximation to:
    10 2001 + 10 2003 10 2002 + 10 2002
  25. Find the pattern and hence calculate:
    1 - 2 + 3 - 4 + 5 - 6 ... + 677 - 678 + ... - 1000
  26. Determine if p = 1 ( x + 2 ) p converges. If it does, then work out what it converges to if:
    1. x = - 5 2
    2. x = - 5
  27. Calculate:        i = 1 5 · 4 - i
  28. The sum of the first p terms of a sequence is p ( p + 1 ) . Find the 10 th term.
  29. The powers of 2 are removed from the set of positive integers
    1 ; 2 ; 3 ; 4 ; 5 ; 6 ; ... ; 1998 ; 1999 ; 2000
    Find the sum of remaining integers.
  30. Observe the pattern below:
    1. If the pattern continues, find the number of letters in the column containing M's.
    2. If the total number of letters in the pattern is 361, which letter will be in the last column.
  31. The following question was asked in a test: Find the value of 2 2005 + 2 2005 . Here are some of the students' answers:
    1. Megan said the answer is 4 2005 .
    2. Stefan wrote down 2 4010 .
    3. Nina thinks it is 2 2006 .
    4. Annatte gave the answer 2 2005 × 2005 .
    Who is correct? (“None of them" is also a possibility.)
  32. A shrub of height 110 cm is planted. At the end of the first year, the shrub is 120 cm tall. Thereafter, the growth of the shrub each year is half of its growth in the previous year. Show that the height of the shrub will never exceed 130 cm.

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Source:  OpenStax, Siyavula textbooks: grade 12 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11242/1.2
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